\documentclass[AutoFakeBold,oneside,12pt]{ctexart}
\usepackage{geometry}

\geometry{a4paper,scale=0.9}

\newcommand{\myind}[1]{{\heiti\upshape\color{black} #1 }\index{#1}}

%%%%%%%% 这两个宏包冲突, 只可保留其中一个
%\usepackage{amsfonts}
\usepackage{mathrsfs}
%%%%%%%%


\usepackage{fancyhdr}
\usepackage{extramarks}
\usepackage{amsmath}
\usepackage{amsthm}

\usepackage{tikz}
\usepackage[plain]{algorithm}
\usepackage{algpseudocode}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{bm}
\usepackage{physics}
\usepackage{calligra}
\usepackage{csquotes}
\usepackage{tensor}
\usepackage[thicklines]{cancel}
\usepackage{tcolorbox}
\usepackage{pstricks}
\usepackage{amssymb}
\usepackage{listings}
\usepackage{enumerate}

\usetikzlibrary{automata,positioning}

%
% Basic Document Settings
%

\topmargin=-0.45in
\evensidemargin=0in
\oddsidemargin=0in
\textwidth=6.5in
\textheight=9.0in
\headsep=0.25in

\linespread{1.1}

\pagestyle{fancy}
\lhead{\hmwkAuthorName}
\chead{\hmwkClass\ \hmwkClassInstructor\ \hmwkClassTime: \hmwkTitle}
\rhead{\firstxmark}
\lfoot{\lastxmark}
\cfoot{\thepage}

\renewcommand\headrulewidth{0.4pt}
\renewcommand\footrulewidth{0.4pt}

\setlength\parindent{0pt}

%
% Create Problem Sections
%

\newcommand{\enterProblemHeader}[1]{
	\nobreak\extramarks{}{习题 \arabic{#1} 接下页\ldots}\nobreak{}
	\nobreak\extramarks{习题 \arabic{#1} (continued)}{习题 \arabic{#1} 接下页\ldots}\nobreak{}
}

\newcommand{\exitProblemHeader}[1]{
	\nobreak\extramarks{习题 \arabic{#1} (continued)}{习题 \arabic{#1} 接下页\ldots}\nobreak{}
	\stepcounter{#1}
	\nobreak\extramarks{习题 \arabic{#1}}{}\nobreak{}
}

\setcounter{secnumdepth}{0}
\newcounter{partCounter}
\newcounter{homeworkProblemCounter}
\setcounter{homeworkProblemCounter}{1}
\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{}

\newenvironment{homeworkProblem}{
	\section{习题 \arabic{homeworkProblemCounter}}
	\setcounter{partCounter}{1}
	\enterProblemHeader{homeworkProblemCounter}
}{
	\exitProblemHeader{homeworkProblemCounter}
}

%
% Homework Details
%   - Title
%   - Due date
%   - Class
%   - Section/Time
%   - Instructor
%   - Author
%

\newcommand{\hmwkTitle}{作业\ \#5}
\newcommand{\hmwkDueDate}{9月22日, 2019}
\newcommand{\hmwkClass}{概率论与随机过程}
\newcommand{\hmwkClassTime}{}
\newcommand{\hmwkClassInstructor}{}
\newcommand{\hmwkAuthorName}{杨勇,2019110294}

%
% Title Page
%

\title{
	\vspace{2in}
	\textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\
	\normalsize\vspace{0.1in}\small{完成于 \hmwkDueDate}\\
	\vspace{0.1in}\large{\textit{\hmwkClassInstructor\ \hmwkClassTime}}
	\vspace{3in}
}

\author{\textbf{\hmwkAuthorName}}
\date{}

\renewcommand{\part}[1]{\textbf{\large Part \Alph{partCounter}}\stepcounter{partCounter}\\}

%
% Various Helper Commands
%

% Useful for algorithms
\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}}

% For derivatives
\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)}

% For partial derivatives
\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)}

% Integral dx
\newcommand{\dx}{\mathrm{d}x}

% Alias for the Solution section header
\newcommand{\solution}{\textbf{\large 解.}}

% Probability commands: Expectation, Variance, Covariance, Bias
\newcommand{\E}{\mathrm{E}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
\newcommand{\Bias}{\mathrm{Bias}}
\newenvironment{blist}%
{\begin{list}{{\hfill\raisebox{1.12pt}{$\bullet$}}}{%
			\leftmargin2em\labelwidth1.5em\labelsep0.5em
			\itemsep0pt\itemindent0pt\parsep0pt\topsep0pt}}
	{\end{list}}
\begin{document}
	
	\maketitle
	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\pagebreak
	\begin{homeworkProblem}
	试证:
	\begin{blist}
		\item[(1)] 若$\xi_n\stackrel{p}{\rightarrow}\xi$, 则$\xi_n-\xi\stackrel{p}{\rightarrow}0$;
		\item[(2)] 若$\xi_n\stackrel{p}{\rightarrow}\xi$, $\xi_n\stackrel{p}{\rightarrow}\eta$,则$P(\xi=\eta) = 1$;
		\item[(3)] 若$\xi_n\stackrel{p}{\rightarrow}\xi$, 则$\xi_n-\xi_m\stackrel{p}{\rightarrow}0$($n,m\to\infty$);
		\item[(4)]
		若$\xi_n\stackrel{p}{\rightarrow}\xi$, 则对任意常数$C$, 有$C\xi_n\stackrel{p}{\rightarrow}C\xi$. 若$\eta$为随机变量, 有$\xi_n\eta\stackrel{p}{\rightarrow}\xi\eta$;
		\item[(5)]
		若$\xi_n\stackrel{p}{\rightarrow}\xi$,$\eta_n\stackrel{p}{\rightarrow}\eta$, 则$\xi_n\pm\eta_n\stackrel{p}{\rightarrow}\xi\pm\eta$, $\xi_n\eta_n\stackrel{p}{\rightarrow}\xi\eta$;
		\item[(6)]
		若$\xi_n\stackrel{p}{\rightarrow}\xi$,$\eta_n\stackrel{p}{\rightarrow}\eta$,且$g(x)$是$\mathbb{R}^{(1)}$上的连续函数, 则$g(\xi_n)\stackrel{p}{\rightarrow}g(\xi)$.
	\end{blist}
	\end{homeworkProblem}
	
	\begin{homeworkProblem}
		设$\{\xi_n\}$是单调下降的正随机变量列, 且$\xi_n\stackrel{p}{\rightarrow}0$,试证$\xi_n\to 0,\mathrm{a.e.}$.
	\end{homeworkProblem}

	\begin{homeworkProblem}
		证明: 若存在常数$C>0$, 使$\abs{\xi_n}<C$, $\abs{\xi}<C$, 则$\xi_n\stackrel{p}{\rightarrow}\xi$的充要条件是$\xi_n$平均收敛于$\xi$.
	\end{homeworkProblem}

\end{document}
